The Role of Drop Shape in Impact Force

Abstract

Drop impacts are ubiquitous in natural and industrial processes, yet the influence of drop shape on impact force remains a fundamental open question. Combining experiments with theoretical analysis, we show that drop shape plays a critical role, with impact force varying by more than an order of magnitude solely due to changes in shape. By uncovering self-similarity in time and cross-shape similarity across diverse drop profiles, we develop a universal cylinder model that accurately predicts both the magnitude and timing of the impact force. This study establishes a comprehensive framework for understanding impact forces across a wide range of drop shapes. Given the prevalence of drop impacts with varying shapes in real-world scenarios, our findings hold fundamental significance and have broad potential applications across industries such as soil erosion prevention, jet cutting, spray coating, and design of windshields and wind turbines.

Figures (3)

• Figure 1: Experimental setup and impact force measurements. (a) The experimental setup for producing non-spherical drops. (b) Cross-sections of drops with various shapes generated by our method. (c) Snapshots of drop impacting on a solid surface by oblate (top), spherical (middle), and prolate (bottom) drops. (d) The impact force (left panel) and its dimensionless form (right panel) for oblate, spherical, and prolate drops measured by force sensor. (e) The dimensionless maximum force $F_{\rm max}^*$ as the function of the aspect ratio $\alpha$. (f) The dimensionless time to reach the maximum force $t_{\rm Fmax}^*$ as the function of $\alpha$.
• Figure 2: The self-similarity and cross-shape similarity of the impact force distribution. (a) The evolution of the force density distribution calculated from pressure gradient along $z$-axis, denoted as $f_p (r,z)$ in the natural coordinates (top), whose rescaled representation $\mathcal{f}_\mathcal{P} (\xi,\eta)$ exhibits self-similar behavior (bottom). (b) The force density distribution $f_p (r,z)$ of different elliptical drops at early stage (top), whose rescaled representation $\mathcal{f}_\mathcal{P} (\xi,\eta)$ exhibits cross-shape similarity across different $\alpha$ (bottom). (c) The force distribution $f_p (r,z)$ of different superelliptical drops at early impact stage (top), whose rescaled representation $n\mathcal{f}_\mathcal{P} (\xi,\eta)$ exhibits cross-shape similarity across different $\alpha$ and $n$. (bottom).
• Figure 3: The cylinder model. (a) The simulation of rescaled force $\mathcal{f}_\mathcal{P}$ for a spherical drop ($n=2$). The high-$\mathcal{f}_\mathcal{P}$ cylindrical region is marked by the red square. (b) The impact force integrated within a cylinder ($\mathcal{F}$) with varying heights ($\tilde{h}$) divided by the total impact force $\mathcal{F}_{\rm tot}$. (c) Schematics illustrating the growth of the cylinder to reach maximum volume for drops with different aspect ratios. (d) Comparison between the cylinder model prediction and experimental data on $t_{\rm Fmax}^*$. The dashed curve represents the elliptical shapes at fixed $n=2$ and the solid curve represents the superelliptical shapes with different $\alpha$ and $n$. (e) The schematics of superellipse demonstrates a flatter drop profile for a larger n across different aspect ratio $\alpha$. (f) The prediction on $F_{\rm max}^*$ by the cylinder model. The dashed curve represents the elliptical shapes with fixed $n=2$ and the solid curve represents the superelliptical shapes with different $\alpha$ and $n$.